• Non-metric (or morphological) vs. metric
• Rogers (1999) introduces new method for sex estimation
  • non-metric, using only the distal humerus
  • 4 traits:
    • Olecranon fossa shape
    • Angle of the medial epicondyle
    • Trochlear Extension / Symmetry
    • Trochlear Constriction
  • 74-94% overall accuracy
  • seems to be somewhat reliable in juveniles (~81% accuracy in Rogers, 2009)
• Falys et al (2005) tested the method
  • 69-82% accuracy
• Vance et al (2011) also tested it. Later they also did a GM version of it (Vance & Steyn, 2013)
  • 74-79% accuracy
  • GM paper: 78-91% accuracy

Rogers (1999); Falys et al (2005).

Rogers (1999); Falys et al (2005).

Rogers (2009)

Vance et al. 2011

Rogers (1999); Falys et al (2005).

• 80 female and 71 male humeri (left side)
• from the 21st Century Identified Skeletal Collection (CEI/XXI)
  • at the University of Coimbra, Portugal.
• All specimens were photographed in standardized position
  • from the posterior view
  • by placing the subject on a flat table
  • and the camera directly above the olecranon fossa.
• With TPSdig2 (Rohlf, 2006) we digitized
  • open curves (20 points)
  • closed outlines (32 points)

The first evaluated the constriction of the trochlea, with the curve reaching from the most medial inferior aspect to the most lateral inferior aspect of the trochlea. The second described the outline of the olecranon fossa and was recorded clockwise starting from its most inferior aspect.

Individual 4 (female) and Individual 6 (male) of the CEI/XXI collection.

• All the analyses were performed using the R programming language (R Core Team 2017)
  • and the Momocs package developed by Bonhomme et al. (2014).
• Generalized Procrustes analysis (GPA) as first proposed by Gower (1975).
• Natural 8 degrees polynomials fits were used to define the open curve of the trochlea
• An elliptical Fourier transform was used for the closed outline of the olecranon fossa
  • 99% of the harmonic power was kept
  • in this case, that represented 6 harmonics

From then on, both sets of data were processed with the same statistical algorithms.

Hereby, both sets of data were processed with the same statistical algorithms.

• Principal component analysis (PCA) to reduce dimensionality, and to visualize the morphospace.
• Linear discriminant analyses (LDA)
  • using the principal components, to estimate the sex from shape configurations
  • trained on a K = n - 1 cross-validation scheme to avoid overfitting
• The method was implemented into a webapp, through the Shiny package (Chang et al, 2017)

This follows the recent trend in forensic anthropology, which attempts to make complicated statistical models more accessible and easier to utilize through interactive online applications (e.g. Navega et al., 2015; Gonçalves et al, 2016; Curate et al, 2016).

TC.n <- npoly(TC.aligned, nb.pts = 16, degree = 8)
TC.pca <- PCA(TC.n)
TC.pca %>% plot(., "Sex", col = palet)

TC.pca %>% as_df() %>% ggplot() + aes(x = PC1, y = PC2, col = Sex) +
    geom_point() + geom_density2d() + theme_light()

TC.ms <- mshapes(TC.n, 1)
TC_female <- TC.ms$shp$Female %T>% coo_plot(border = palet[1])
TC_male <- TC.ms$shp$Male %T>% coo_draw(border = palet[2])
legend("topright", lwd = 1, col = palet, legend = c("Female", "Male"), cex = 0.8)

tps_grid(TC_female, TC_male, shp.border = palet)

tps_arr(TC_female, TC_male, shp.border = palet)

tps_iso(TC_female, TC_male, shp.border = palet)

MANOVA(TC.pca, 1)

## PC axes 1 to 2 were retained

##            Df Hotelling-Lawley approx F num Df den Df  Pr(>F)  
## fac         1         0.031687   2.3448      2    148 0.09942 .
## Residuals 149                                                  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

TC.lda <- LDA(TC.pca, 1)

## 0.99 total variance

## 2 PC retained

TC.lda

##  * Leave-one-out cross-validation ($CV.correct): (60.9% - 92/151): 
## 
##  * Class correctness ($CV.ce):
##    Female      Male 
## 0.7875000 0.4084507 
## 
##  * Cross-validation table ($CV.tab):
##         classified
## actual   Female Male
##   Female     63   17
##   Male       42   29

plot(TC.lda, palette = cores2)

## NULL

OF.ef <- efourier(OF.aligned, nb.h = 6)
OF.pca <- PCA(OF.ef)
OF.pca %>% plot(., "Sex", col = palet)

OF.pca %>% as_df() %>% ggplot() + aes(x = PC1, y = PC2, col = Sex) + coord_equal() + 
    geom_point() + geom_density2d() + theme_light()

OF.ms <- mshapes(OF.ef, 1)
OF_female <- OF.ms$shp$Female %T>% coo_plot(border = palet[1])
OF_male <- OF.ms$shp$Male %T>% coo_draw(border = palet[2])
legend("topright", lwd = 1, col = palet, legend = c("Female", "Male"), cex = 0.8)

tps_grid(OF_female, OF_male, shp.border = palet)

tps_arr(OF_female, OF_male, shp.border = palet)

tps_iso(OF_female, OF_male, grid = TRUE, shp.border = palet)

MANOVA(OF.pca, 1)

## PC axes 1 to 7 were retained

##            Df Hotelling-Lawley approx F num Df den Df    Pr(>F)    
## fac         1           1.8666   38.133      7    143 < 2.2e-16 ***
## Residuals 149                                                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

OF.lda <- LDA(OF.pca, 1)

## 0.99 total variance

## 7 PC retained

OF.lda

##  * Leave-one-out cross-validation ($CV.correct): (92.1% - 139/151): 
## 
##  * Class correctness ($CV.ce):
##    Female      Male 
## 0.9500000 0.8873239 
## 
##  * Cross-validation table ($CV.tab):
##         classified
## actual   Female Male
##   Female     76    4
##   Male        8   63

plot(OF.lda, palette = cores2)

## NULL

By better we mean:

• as simple as possible
• keeping similar levels of overall accuracy
• applicable!!!

cor(OF.pca$x[,1:7], as.numeric(OF.pca$fac$Sex)) # correlations

##            [,1]
## PC1  0.05302026
## PC2  0.46336920
## PC3 -0.11872418
## PC4  0.63185511
## PC5 -0.09388462
## PC6  0.05482801
## PC7  0.09208552

Let's see if it makes sense. Little exclusive for Morph2017

OF.pca %>% as_df() %>% ggplot() + aes(x = PC2, y = PC4, col = Sex) + 
    geom_point() + geom_density2d() + theme_light()

OF.lda2 <- LDA(OF.pca$x[,c(2,4)], OF.pca$fac$Sex)
OF.lda2

##  * Leave-one-out cross-validation ($CV.correct): (91.4% - 138/151): 
## 
##  * Class correctness ($CV.ce):
##    Female      Male 
## 0.9375000 0.8873239 
## 
##  * Cross-validation table ($CV.tab):
##         classified
## actual   Female Male
##   Female     75    5
##   Male        8   63

This has been implemented in an interactive, easy-to-use, free to all, way:

• Check our web-app!